On a class of pseudodifferential operators with mixed homogeneities

Transcription

1 On a class of pseudodifferential operators with mixed homogeneities Po-Lam Yung University of Oxford July 25, 2014

2 Introduction Joint work with E. Stein (and an outgrowth of work of Nagel-Ricci-Stein-Wainger, to appear) Motivating question: what happens when one compose two operators of two different homogeneities? e.g. On R N one can associate two different dilations: for x = (x, x N ) R N, λ > 0, one can define λ x := (λx, λx N ) λ x := (λx, λ 2 x N ) (isotropic) (non-isotropic) Associated to these are two norms, each homogeneous with respect to one of these dilations: x = x + x N x = x + x N 1/2

3 There are also the dual norms, on the cotangent space of R N, given by ξ = ξ + ξ N ξ = ξ + ξ N 1/2 One could look at multipliers e(ξ), with α ξ β ξ N e(ξ) ξ α β, and their associated multiplier operators T e f (ξ) = e(ξ) f (ξ). (These are just standard isotropic singular integral operators.) One could also look at multipliers h(ξ), with α ξ β ξ N h(ξ) ξ α 2 β, and their associated multiplier operators T h f (ξ) = h(ξ) f (ξ). (These are just non-isotropic singular integral operators, arising e.g. when one solves the heat equation.)

4 Some motivating Questions What happens when one compose T e with T h? What kind of operator do we get? What are the nature of the singularities of the multiplier, or the kernel, of T e T h? What mapping properties does T e T h satisfy? When is it (say) weak-type (1,1)? The question of composition is quite easy, since we are dealing with convolution operators on an abelian group R N. The question of mapping properties was already studied in a paper of Phong and Stein in We are interested in these questions, because they serve as toy model problems for what one needs to do in more general settings.

5 e.g. In several complex variables, in solving the -Neumann problem on a smooth strongly pseudoconvex domain in C n+1, one faces the problem of inverting the Calderon operator +. Roughly speaking, this amounts to composing an operator with isotropic homogeneity, with an operator with non-isotropic homogeneity: b (at least when n > 1; c.f. Greiner-Stein 1977). Similar compositions arise in the study of the Hodge Laplacian on k-forms on the Heisenberg group H n (Müller-Peloso-Ricci 2012)

6 The role of flag kernels The flag kernels are integral kernels that are singular along certain subspaces on R N (e.g. Nagel-Ricci-Stein 2001, Nagel-Ricci-Stein-Wainger 2011, to appear). These are special cases of product kernels, which were studied by many authors (e.g. R. Fefferman-Stein 1982, Journe 1985, Nagel-Stein 2004). e.g. On the Heisenberg group H n, a flag kernel could be singular along the t-axis, and satisfies K(z, t) z 2n ( z 2 + t ) 1 along with some corresponding differential inequalities and cancellation conditions.

7 Simple examples of flag kernels include both singular integral kernels with isotropic homogeneities, and those with non-isotropic homogeneities. More sophisticated examples of flag kernels on H n are given by the joint spectral multipliers m(l 0, it ), where m is a Marcinkiewicz multiplier, and L 0 is the sub-laplacian (Müller-Ricci-Stein 1995, 1996). The singular integral operators with flag kernels map L p to L p, for 1 < p <, and form an algebra under composition. Thus if we want to compose a singular integral with isotropic homogeneity, with one that has non-isotropic homogeneity, we could have composed them in the class of all flag kernels. But then we get as a result a flag kernel, which is singular along some subspaces (whereas our original kernels are both singular only at one point).

8 It turns out one should consider the intersection of those flag kernels that are singular along the t-axis, with those that are singular along the z-axis, as in Nagel-Ricci-Stein-Wainger (to appear). This gives rise to operators with mixed homogeneities: K(z, t) ( z + t ) 2n ( z 2 + t ) 1 along with differential inequalities and cancellation conditions. Our first result will be a pseudodifferential realization of the above operators of with mixed homogeneities. The goal is to write them as pseudodifferential operators: ˆ T a f (x) = a(x, ξ)ˆf (ξ)e 2πix ξ dξ H n for some suitable symbols a(x, ξ).

9 Nagel-Stein (1979) and Beals-Greiner (1988) has realized those singular integrals with purely non-isotropic homogeneities as pseudodifferential operators. We will do so for singular integrals with mixed homogeneities; in doing so, we will also consider operators of all orders (not just order 0 ones, as singular integrals would be). Our results will actually hold in a more general setting, outside several complex variables; it will hold as long as a smooth distribution of tangent subspaces (of constant rank) is given on R N. We will also see some geometric invariance of our class of operators as we proceed. A very step-2 theory!

10 Our set-up Suppose on R N, we are given a (global) frame of tangent vectors, namely X 1,..., X N, with X i = N j=1 Aj i (x). x j We assume that all A j i (x) are C functions, and that J x A j i (x) L (R N ) for all multiindices J. We also assume that det(a j i (x)) is uniformly bounded from below on R N. Let D be the distribution of tangent subspaces on R N given by the span of {X 1,..., X N 1 }. Our constructions below seem to depend on the choice of the frame X 1,..., X N, but ultimately the class of operators we introduce will only depend on D. No curvature assumption on D is necessary!

11 An example: the contact distribution on H 1 (Picture courtesy of Assaf Naor)

12 Geometry of the distribution We write θ 1,..., θ N for the frame of cotangent vectors dual to X 1,..., X N. We will need a variable seminorm ρ x (ξ) on the cotangent bundle of R N, defined as follows. Given ξ = N i=1 ξ idx i, and a point x R N, we write ξ = N (M x ξ) i θ i. i=1 Then ρ x (ξ) := N 1 i=1 (M x ξ) i. We also write ξ = N i=1 ξ i for the Euclidean norm of ξ.

13 The variable seminorm ρ x (ξ) induces a quasi-metric d(x, y) on R N. If x, y R N with x y < 1, we write { } 1 d(x, y) := sup : (x y) ξ = 1. ρ x (ξ) + ξ 1/2 We also write x y for the Euclidean distance between x and y.

14 Our class of symbols with mixed homogeneities The symbols we consider will be assigned two different orders, namely m and n, which we think of as the isotropic and non-isotropic orders of the symbol respectively. In the case of the constant distribution, it is quite easy to define the class of symbols we are interested in: given m, n R, if a 0 (x, ξ) C (T R N ) is such that where a 0 (x, ξ) (1 + ξ ) m (1 + ξ ) n J x α ξ β ξ N a 0 (x, ξ) J,α,β (1 + ξ ) m β (1 + ξ ) n α ξ = ξ + ξ N 1/2 if ξ = N ξ i dx i, i=1 then we say a 0 S m,n (D 0 ). (Note that in this situation, 1 + ξ 1 + ρ x (ξ) + ξ 1/2.)

15 More generally, suppose D is a distribution as before (in particular, we fix the frame X 1,..., X N, and its dual frame θ 1,..., θ N ). Given x R N and ξ = N ξ i dx i = i=1 N (M x ξ) i θ i, i=1 write N M x ξ = (M x ξ) i dx i. i=1 Then we say a S m,n (D), if and only if a(x, ξ) = a 0 (x, M x ξ) for some a 0 S m,n (D 0 ).

16 e.g. Suppose we are on R 3. Pick coordinates x = (x 1, x 2, t), and dual coordinates ξ = (ξ 1, ξ 2, τ). Define X 1 = and x 1 + 2x 2 t, X 2 = x 2 2x 1 t, X 3 = t, D := span{x 1, X 2 }. Then identifying R 3 with the Heisenberg group H 1, D is sometimes called the contact distribution. We then have a S m,n (D), if and only if a(x 1, x 2, t, ξ 1, ξ 2, τ) = a 0 (x 1, x 2, t, ξ 1 + 2x 2 τ, ξ 2 2x 1 τ, τ) for some a 0 S m,n (D 0 ). The condition a S m,n (D) can also be phrased in terms of suitable differential inequalities.

17 e.g. Still consider the contact distribution D on H 1. Recall coordinates x = (x 1, x 2, t) on H 1, and dual coordinates ξ = (ξ 1, ξ 2, τ). Let and D 1 = D τ = τ 2x 2 + 2x 1, ξ 1 ξ 2 x 1 + 2τ, D 2 = ξ 2 x 2 2τ, D 3 = ξ 1 t. Then a S m,n (D), if and only if a(x, ξ) C (H 1 R 3 ) satisfies a(x, ξ) (1 + ξ ) m (1 + ρ x (ξ) + ξ 1/2 ) n, α ξ Dβ ξ D Ja(x, ξ) (1 + ξ ) m β (1 + ρ x (ξ) + ξ 1/2 ) n α, where D J is composition of any number of the D 1, D 2, D 3.

18 Back to the general set up. We call elements of S m,n (D) a symbol of order (m, n). Our class of symbols S m,n is quite big. S m,0 contains every standard (isotropic) symbol of order m: a(x, ξ) (1 + ξ ) m α ξ J x a(x, ξ) α,j (1 + ξ ) m α Also, S 0,n contains the following class of symbols, which we think of as non-isotropic symbols of order n: a(x, ξ) (1 + ρ x (ξ) + ξ 1/2 ) n α ξ Dβ ξ DJ a(x, ξ) α,β,j (1 + ρ x (ξ) + ξ 1/2 ) n α 2β Many of the results below for S m,n has an (easier) counter-part for these non-isotropic symbols.

19 Our class of pseudodifferential operators with mixed homogeneities To each symbol a S m,n (D), we associate a pseudodifferential operator ˆ T a f (x) = a(x, ξ) f (ξ)e 2πix ξ dξ. R N We denote the set of all such operators Ψ m,n (D). We remark that Ψ m,n (D) depends only on the distribution D, and not on the choice of the vector fields X 1,..., X N (nor on the choice of a coordinate system on R N ). (Geometric invariance!) One sees that Ψ m,n (D) is the correct class of operators to study, via the following kernel estimates:

20 Theorem (Stein-Y. 2013) If T Ψ m,n (D) with m > 1 and n > (N 1), then one can write ˆ Tf (x) = f (y)k(x, y)dy, R N where the kernel K(x, y) satisfies K(x, y) x y (N 1+n) d(x, y) 2(1+m), (X ) γ x,y δ x,y K(x, y) x y (N 1+n+ γ ) d(x, y) 2(1+m+ δ ). Here X refers to any of the good vector fields X 1,..., X N 1 that are tangent to D, and the subscripts x, y indicates that the derivatives can act on either the x or y variables. c.f. two-flag kernels of Nagel-Ricci-Stein-Wainger

21 Main theorems Theorem (Stein-Y. 2013) If T 1 Ψ m,n (D) and T 2 Ψ m,n (D), then T 1 T 2 Ψ m+m,n+n (D). Furthermore, if T 1 is the adjoint of T 1 with respect to the standard L 2 inner product on R N, then we also have T 1 Ψ m,n (D). In particular, the class of operators Ψ 0,0 form an algebra under composition, and is closed under taking adjoints. Proof unified by introducing some suitable compound symbols.

22 Theorem (Stein-Y. 2013) If T Ψ 0,0 (D), then T maps L p (R N ) into itself for all 1 < p <. Operators in Ψ 0,0 are operators of type (1/2, 1/2). As such they are bounded on L 2. But operators in Ψ 0,0 may not be of weak-type (1,1); this forbids one to run the Calderon-Zygmund paradigm in proving L p boundedness use Littlewood-Paley projections instead, and need to introduce some new strong maximal functions. Nonetheless, there are two very special ideals of operators inside Ψ 0,0, namely Ψ ε, 2ε and Ψ ε,ε for ε > 0. (The fact that these are ideals of the algebra Ψ 0,0 follows from the earlier theorem). They satisfy:

23 Theorem (Stein-Y. 2013) If T Ψ ε, 2ε or Ψ ε,ε for some ε > 0, then (a) T is of weak-type (1,1), and (b) T maps the Hölder space Λ α (R N ) into itself for all α > 0. An analogous theorem holds for some non-isotropic Hölder spaces Γ α (R N ). Proof of (a) by kernel estimates Proof of (b) by Littlewood-Paley characterization of the Hölder spaces Λ α.

24 Theorem (Stein-Y. 2013) Suppose T Ψ m,n with m > 1, n > (N 1), m + n 0, and 2m + n 0. For p 1, define an exponent p by 1 p := 1 { } m + n p γ, 2m + n γ := min, N N + 1 if 1/p > γ. Then: (i) T : L p L p (ii) if m+n N for 1 < p p < ; and 2m+n N+1, then T is weak-type (1, 1 ). These estimates for S m,n are better than those obtained by composing between the optimal results for S 0,n and S m,0.

25 For example, take the example of the 1-dimensional Heisenberg group H 1 (so N = 3). If T 1 is a standard (or isotropic) pseudodifferential operator of order 1/2 (so T 1 Ψ 1/2,0 ), and T 2 is a non-isotropic pseudodifferential operator of order 1 (so T 2 Ψ 0, 1 ), then the best we could say about the operators individually are just T 2 : L 4/3 L 2, T 1 : L 2 L 3. But according to the previous theorem, which is better. T 1 T 2 : L 4/3 L 4, Proof of (i) by interpolation between Ψ 0,0 and Ψ 1, (N 1). Proof of (ii) by kernel estimates.

26 Some applications One can localize the above theory of operators of class Ψ m,n to compact manifolds without boundary. Let M = Ω be the boundary of a strongly pseudoconvex domain Ω in C n+1, n 1. Then the Szegö projection S is an operator in Ψ ε, 2ε for all ε > 0; c.f. Phong-Stein (1977). Furthermore, the Dirichlet-to- -Neumann operator +, which one needs to invert in solving the -Neumann problem on Ω, has a parametrix in Ψ 1, 2 ; c.f. Greiner-Stein (1977), Chang-Nagel-Stein (1992).

27 Epilogue On R, we know that if Tf := f y 1+α, 0 < α < 1, then whenever Tf L q C f L p, 1 q = 1 α, 1 < p < q <. p There are many known proofs; below we sketch one that is perhaps less commonly known, and that is reminiscent of our proof of the smoothing properties for our class Ψ m,n above.

28 The idea is to use complex interpolation: For s C with 0 Re s 1, let k s be the tempered distribution k s (y) = (1 + s) 1 2 y y 1+s, so that when 0 < Re s 1, we have k s (y) = s y 1+s. Let T s f = f k s for f S. Then when Re s = 1, we have T s : L 1 L, since it is then a convolution against a bounded function. Also when Re s = 0, we have T s : L r L r for all 1 < r <, since k s is then a Calderon-Zygmund kernel (one can check that k s L for such s, since k s, being a derivative, satisfies a cancellation condition). Interpolation then shows that T α : L p L q, when 1 q = 1 α, 1 < p < q <. p